Differential Equations Question 4
Question 4 - 2024 (01 Feb Shift 2)
If $\frac{d x}{d y}=\frac{1+x-y^{2}}{y}, x(1)=1$, then $5 x(2)$ is equal to :
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Answer (5)
Solution
$\frac{d x}{d y}-\frac{x}{y}=\frac{1-y^{2}}{y}$
Integrating factor $=e^{\int-\frac{1}{y} d y}=\frac{1}{y}$
$x \cdot \frac{1}{y}=\int \frac{1-y^{2}}{y^{2}} d y$
$\frac{x}{y}=\frac{-1}{y}-y+c$
$x=-1-y^{2}+c y$
$x(1)=1$
$1=-1-1+c \Rightarrow c=3$
$x=-1-y^{2}+3 y$
$5 x(2)=5(-1-4+6)$
$=5$
